³¹P NMR: A Spin-½, 100% Abundant Two-Level Quantum System — E8 Intelligence Research

FINDING: ³¹P is a spin-½ NMR-active nucleus (I = ½, 100% natural abundance), making it a clean two-level quantum system for magnetic resonance. MATH: - Nuclear spin quantum number: I = ½ → multiplicity 2I+1 = 2 (two Zeeman eigenstates m_I = ±½) - Composition: Z = 15 protons, N = 16 neutrons, A = 31 (odd A → half-integer spin) - Larmor frequency: ω₀ = γB₀, with ³¹P gyromagnetic ratio γ ≈ 1.08 × 10⁸ rad·s⁻¹·T⁻¹ (17.235 MHz/T) - Zeeman splitting: ΔE = γℏB₀ - NMR transition (single-quantum): Δm_I = ±1 - Double-quantum coherence: Δm_I = ±2 (referenced in findings for spin pairs / quadrupolar nuclei) - Schrödinger eigenvalue problem: Ĥψ = Eψ governs the spin state evolution - Magnetization precession: dM/dt = γM × B (Bloch equation) CONNECTION: - Spin-½ is the fundamental representation of SU(2); the two-level system maps to the Bloch sphere — a geometric object with the golden-ratio-free but maximally symmetric structure of CP¹. - SU(2) ≅ unit quaternions; double cover of SO(3). The 2:1 c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com